from sympy.vector.basisdependent import (BasisDependent, BasisDependentAdd,
BasisDependentMul, BasisDependentZero)
from sympy.core import S, Pow
from sympy.core.expr import AtomicExpr
from sympy import ImmutableMatrix as Matrix
import sympy.vector
[docs]class Dyadic(BasisDependent):
"""
Super class for all Dyadic-classes.
References
==========
.. [1] https://en.wikipedia.org/wiki/Dyadic_tensor
.. [2] Kane, T., Levinson, D. Dynamics Theory and Applications. 1985
McGraw-Hill
"""
_op_priority = 13.0
@property
def components(self):
"""
Returns the components of this dyadic in the form of a
Python dictionary mapping BaseDyadic instances to the
corresponding measure numbers.
"""
# The '_components' attribute is defined according to the
# subclass of Dyadic the instance belongs to.
return self._components
[docs] def dot(self, other):
"""
Returns the dot product(also called inner product) of this
Dyadic, with another Dyadic or Vector.
If 'other' is a Dyadic, this returns a Dyadic. Else, it returns
a Vector (unless an error is encountered).
Parameters
==========
other : Dyadic/Vector
The other Dyadic or Vector to take the inner product with
Examples
========
>>> from sympy.vector import CoordSys3D
>>> N = CoordSys3D('N')
>>> D1 = N.i.outer(N.j)
>>> D2 = N.j.outer(N.j)
>>> D1.dot(D2)
(N.i|N.j)
>>> D1.dot(N.j)
N.i
"""
Vector = sympy.vector.Vector
if isinstance(other, BasisDependentZero):
return Vector.zero
elif isinstance(other, Vector):
outvec = Vector.zero
for k, v in self.components.items():
vect_dot = k.args[1].dot(other)
outvec += vect_dot * v * k.args[0]
return outvec
elif isinstance(other, Dyadic):
outdyad = Dyadic.zero
for k1, v1 in self.components.items():
for k2, v2 in other.components.items():
vect_dot = k1.args[1].dot(k2.args[0])
outer_product = k1.args[0].outer(k2.args[1])
outdyad += vect_dot * v1 * v2 * outer_product
return outdyad
else:
raise TypeError("Inner product is not defined for " +
str(type(other)) + " and Dyadics.")
def __and__(self, other):
return self.dot(other)
__and__.__doc__ = dot.__doc__
[docs] def cross(self, other):
"""
Returns the cross product between this Dyadic, and a Vector, as a
Vector instance.
Parameters
==========
other : Vector
The Vector that we are crossing this Dyadic with
Examples
========
>>> from sympy.vector import CoordSys3D
>>> N = CoordSys3D('N')
>>> d = N.i.outer(N.i)
>>> d.cross(N.j)
(N.i|N.k)
"""
Vector = sympy.vector.Vector
if other == Vector.zero:
return Dyadic.zero
elif isinstance(other, Vector):
outdyad = Dyadic.zero
for k, v in self.components.items():
cross_product = k.args[1].cross(other)
outer = k.args[0].outer(cross_product)
outdyad += v * outer
return outdyad
else:
raise TypeError(str(type(other)) + " not supported for " +
"cross with dyadics")
def __xor__(self, other):
return self.cross(other)
__xor__.__doc__ = cross.__doc__
[docs] def to_matrix(self, system, second_system=None):
"""
Returns the matrix form of the dyadic with respect to one or two
coordinate systems.
Parameters
==========
system : CoordSys3D
The coordinate system that the rows and columns of the matrix
correspond to. If a second system is provided, this
only corresponds to the rows of the matrix.
second_system : CoordSys3D, optional, default=None
The coordinate system that the columns of the matrix correspond
to.
Examples
========
>>> from sympy.vector import CoordSys3D
>>> N = CoordSys3D('N')
>>> v = N.i + 2*N.j
>>> d = v.outer(N.i)
>>> d.to_matrix(N)
Matrix([
[1, 0, 0],
[2, 0, 0],
[0, 0, 0]])
>>> from sympy import Symbol
>>> q = Symbol('q')
>>> P = N.orient_new_axis('P', q, N.k)
>>> d.to_matrix(N, P)
Matrix([
[ cos(q), -sin(q), 0],
[2*cos(q), -2*sin(q), 0],
[ 0, 0, 0]])
"""
if second_system is None:
second_system = system
return Matrix([i.dot(self).dot(j) for i in system for j in
second_system]).reshape(3, 3)
class BaseDyadic(Dyadic, AtomicExpr):
"""
Class to denote a base dyadic tensor component.
"""
def __new__(cls, vector1, vector2):
Vector = sympy.vector.Vector
BaseVector = sympy.vector.BaseVector
VectorZero = sympy.vector.VectorZero
# Verify arguments
if not isinstance(vector1, (BaseVector, VectorZero)) or \
not isinstance(vector2, (BaseVector, VectorZero)):
raise TypeError("BaseDyadic cannot be composed of non-base " +
"vectors")
# Handle special case of zero vector
elif vector1 == Vector.zero or vector2 == Vector.zero:
return Dyadic.zero
# Initialize instance
obj = super(BaseDyadic, cls).__new__(cls, vector1, vector2)
obj._base_instance = obj
obj._measure_number = 1
obj._components = {obj: S(1)}
obj._sys = vector1._sys
obj._pretty_form = (u'(' + vector1._pretty_form + '|' +
vector2._pretty_form + ')')
obj._latex_form = ('(' + vector1._latex_form + "{|}" +
vector2._latex_form + ')')
return obj
def __str__(self, printer=None):
return "(" + str(self.args[0]) + "|" + str(self.args[1]) + ")"
_sympystr = __str__
_sympyrepr = _sympystr
class DyadicMul(BasisDependentMul, Dyadic):
""" Products of scalars and BaseDyadics """
def __new__(cls, *args, **options):
obj = BasisDependentMul.__new__(cls, *args, **options)
return obj
@property
def base_dyadic(self):
""" The BaseDyadic involved in the product. """
return self._base_instance
@property
def measure_number(self):
""" The scalar expression involved in the definition of
this DyadicMul.
"""
return self._measure_number
class DyadicAdd(BasisDependentAdd, Dyadic):
""" Class to hold dyadic sums """
def __new__(cls, *args, **options):
obj = BasisDependentAdd.__new__(cls, *args, **options)
return obj
def __str__(self, printer=None):
ret_str = ''
items = list(self.components.items())
items.sort(key=lambda x: x[0].__str__())
for k, v in items:
temp_dyad = k * v
ret_str += temp_dyad.__str__(printer) + " + "
return ret_str[:-3]
__repr__ = __str__
_sympystr = __str__
class DyadicZero(BasisDependentZero, Dyadic):
"""
Class to denote a zero dyadic
"""
_op_priority = 13.1
_pretty_form = u'(0|0)'
_latex_form = r'(\mathbf{\hat{0}}|\mathbf{\hat{0}})'
def __new__(cls):
obj = BasisDependentZero.__new__(cls)
return obj
def _dyad_div(one, other):
""" Helper for division involving dyadics """
if isinstance(one, Dyadic) and isinstance(other, Dyadic):
raise TypeError("Cannot divide two dyadics")
elif isinstance(one, Dyadic):
return DyadicMul(one, Pow(other, S.NegativeOne))
else:
raise TypeError("Cannot divide by a dyadic")
Dyadic._expr_type = Dyadic
Dyadic._mul_func = DyadicMul
Dyadic._add_func = DyadicAdd
Dyadic._zero_func = DyadicZero
Dyadic._base_func = BaseDyadic
Dyadic._div_helper = _dyad_div
Dyadic.zero = DyadicZero()
